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If the rank of the matrix $\left(\begin{array}{ccc}-1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1\end{array}\right)$ is 1, then the value of a is
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Verified Answer
The correct answer is:
$-6$
Let
$\mathrm{A}=\left|\begin{array}{ccc}
-1 & 2 & 5 \\
2 & -4 & \mathrm{a}-4 \\
1 & -2 & \mathrm{a}+1
\end{array}\right|-\left|\begin{array}{ccc}
-1 & 2 & 5 \\
0 & 0 & \mathrm{a}+6 \\
0 & 0 & \mathrm{a}+6
\end{array}\right|$
$\left[\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}+2 \mathrm{R}_{1}, \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}+\mathrm{R}_{1}\right]$
clearly rank of $A$ is 1 if $a=-6$
$\mathrm{A}=\left|\begin{array}{ccc}
-1 & 2 & 5 \\
2 & -4 & \mathrm{a}-4 \\
1 & -2 & \mathrm{a}+1
\end{array}\right|-\left|\begin{array}{ccc}
-1 & 2 & 5 \\
0 & 0 & \mathrm{a}+6 \\
0 & 0 & \mathrm{a}+6
\end{array}\right|$
$\left[\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}+2 \mathrm{R}_{1}, \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}+\mathrm{R}_{1}\right]$
clearly rank of $A$ is 1 if $a=-6$
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